Random Constraint Satisfaction Problems exhibit several phase transitions when their density of constraints is varied. One of these threshold phenomena, known as the clustering or dynamic transition, corresponds to a transition for an information theoretic problem called tree reconstruction. In this article we study this threshold for two CSPs, namely the bicoloring of $k$-uniform hypergraphs with a density $\alpha$ of constraints, and the $q$-coloring of random graphs with average degree $c$. We show that in the large $k,q$ limit the clustering transition occurs for $\alpha = \frac{2^{k-1}}{k} (\ln k + \ln \ln k + \gamma_{\rm d} + o(1))$, $c= q (\ln q + \ln \ln q + \gamma_{\rm d}+ o(1))$, where $\gamma_{\rm d}$ is the same constant for both models. We characterize $\gamma_{\rm d}$ via a functional equation, solve the latter numerically to estimate $\gamma_{\rm d} \approx 0.871$, and obtain an analytic lowerbound $\gamma_{\rm d} \ge 1 + \ln (2 (\sqrt{2}-1)) \approx 0.812$. Our analysis unveils a subtle interplay of the clustering transition with the rigidity (naive reconstruction) threshold that occurs on the same asymptotic scale at $\gamma_{\rm r}=1$.

中文翻译：

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随机约束满足问题聚类转移的渐近性

随机约束满足问题在约束密度变化时表现出几种相变。这些阈值现象之一，称为聚类或动态转换，对应于称为树重建的信息论问题的转换。在本文中，我们研究了两个 CSP 的阈值，即具有约束密度 $\alpha$ 的 $k$-均匀超图的双色，以及平均度为 $c$ 的随机图的 $q$-着色。我们表明，在较大的 $k,q$ 限制中，聚类转换发生在 $\alpha = \frac{2^{k-1}}{k} (\ln k + \ln \ln k + \gamma_{\ rm d} + o(1))$, $c= q (\ln q + \ln \ln q + \gamma_{\rm d}+ o(1))$，其中 $\gamma_{\rm d} $ 是两个模型的相同常数。我们通过函数方程表征 $\gamma_{\rm d}$，对后者进行数值求解以估计 $\gamma_{\rm d} \approx 0.871$，并获得解析下界 $\gamma_{\rm d} \ge 1 + \ln (2 (\sqrt{2}-1)) \约 0.812$。我们的分析揭示了聚类转换与刚性（朴素重建）阈值之间的微妙相互作用，该阈值发生在 $\gamma_{\rm r}=1$ 的相同渐近尺度上。